基于矩阵-向量乘积的近最优层次矩阵逼近
Near-optimal hierarchical matrix approximation from matrix-vector products
AI总结：
提出仅用 A 与 A^T 矩阵-向量乘积构造近最优 HODLR 逼近的随机剥离算法，给出查询与计算复杂度上下界及低秩扰动稳定性保证。
摘要：
我们描述了一种随机算法，用于为仅能通过与 A 和 A^T 的矩阵-向量乘积访问的 n×n 矩阵 A 生成近最优的层次非对角低秩（HODLR）逼近。我们证明，对于秩为 k 的 HODLR 逼近问题，该方法使用 O(k log(n)/β^3) 次矩阵-向量乘积，在期望 Frobenius 范数下达到 (1+β)^{log(n)}-最优逼近。具体而言，该算法使用 O(k log^4(n)/ε^3) 次矩阵-向量乘积获得 (1+ε)-最优逼近；并且对于任意常数 c，使用 O(k log(n)) 次矩阵-向量乘积获得 n^c-最优逼近。除矩阵-向量乘积外，该方法的额外计算成本仅为 O(n poly(log(n), k, β))。我们以上界相配套给出一个下界，表明任何矩阵-向量查询算法若要获得 (1+ε)-最优逼近，至少需要 Ω(k log(n)+k/ε) 次查询。我们的算法可视为广泛用于恢复 HODLR 矩阵的“剥离”方法的鲁棒版本；据我们所知，它是首个在任意层次矩阵类的逼近方面享有理论最坏情况保证的矩阵-向量查询算法。为控制层次逼近各层之间的误差传播，我们提出了一种新的低秩逼近扰动界，表明广泛使用的 Generalized Nyström 方法在以含噪矩阵-向量乘积实现时具有内在稳定性。我们还提出了一种新颖的随机穿孔矩阵草图方法，以进一步控制剥离算法中的误差。
英文摘要：
We describe a randomized algorithm for producing a near-optimal hierarchical off-diagonal low-rank (HODLR) approximation to an $n\times n$ matrix $\mathbf{A}$, accessible only though matrix-vector products with $\mathbf{A}$ and $\mathbf{A}^{\mathsf{T}}$. We prove that, for the rank-$k$ HODLR approximation problem, our method achieves a $(1+β)^{\log(n)}$-optimal approximation in expected Frobenius norm using $O(k\log(n)/β^3)$ matrix-vector products. In particular, the algorithm obtains a $(1+\varepsilon)$-optimal approximation with $O(k\log^4(n)/\varepsilon^3)$ matrix-vector products, and for any constant $c$, an $n^c$-optimal approximation with $O(k \log(n))$ matrix-vector products. Apart from matrix-vector products, the additional computational cost of our method is just $O(n \operatorname{poly}(\log(n), k, β))$. We complement the upper bound with a lower bound, which shows that any matrix-vector query algorithm requires at least $Ω(k\log(n) + k/\varepsilon)$ queries to obtain a $(1+\varepsilon)$-optimal approximation. Our algorithm can be viewed as a robust version of widely used "peeling" methods for recovering HODLR matrices and is, to the best of our knowledge, the first matrix-vector query algorithm to enjoy theoretical worst-case guarantees for approximation by any hierarchical matrix class. To control the propagation of error between levels of hierarchical approximation, we introduce a new perturbation bound for low-rank approximation, which shows that the widely used Generalized Nyström method enjoys inherent stability when implemented with noisy matrix-vector products. We also introduce a novel randomly perforated matrix sketching method to further control the error in the peeling algorithm.
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